Rayleigh Model

The major isotopic findings are summarized in Figure 8. Amean 8 Mo offset of 1.8%o was observed between dissolved and oxide-bound Mo in all experiments. The data do not fit the expectations of a Rayleigh model, as might be expected from irreversible adsorption. Instead, the data closely approximate closed system exchange between adsorbed and dissolved Mo, suggesting that Mo isotope fractionation in this system is an equilibrium process. 

This is valid under relatively small signal excitation conditions, and describes the motion of domain walls in local random fields, a describes the irreversibility of the domain wall motion. Under the conditions where the Rayleigh model holds, the hysteresis in the piezoelectric... 

The Rayleigh model was developed for a single stage batch distillation-where a liquid mixture is charged in a still and a vapour is produced by heating the liquid. At any time, the vapour on top of the liquid is in equilibrium with the liquid left in the still. The vapour is removed as soon as it is produced but no part of the vapour is returned as reflux to the still after condensation. 

Starting with an initial amount of liquid (B0) with composition xB0 charged in the still, the amount (Bt) of liquid left in the still and its composition (xBI) at any time can be calculated using the Rayleigh model (Equation 1.1). 

Table 4.1. Distillation Profile Using The Rayleigh Model...

The tr ectories for denitrification, nitrate uptake, and sedimentary denitrification were each calculated using a closed-system Rayleigh model with an isotope discrimination factor ( ) in the middle of the range given in Rg. 29.2. The two nitrification tr ectories shown were calculated with a mass balance model for remineralization and nitrification of organic matter produced from average deep water N03, or diazotrophic organic matter with a of —2%o (Montoya etal,2m2). 

If no liquid is kept within the cloud ( l = 0), these equations correspond to the classical Rayleigh model, and to a closed system if all liquid is kept. The isotopic composition of the precipitation can be calculated all along the condensation process, if the amount of water leaving the cloud at each step is known. The largest range of 5-variation corresponds to the immediate removal of the condensate (Rayleigh process), and there is a decrease of this range by about a factor of 2, when all liquid formed is kept in the cloud (Jouzel, 1986). 

A Rayleigh model (Dansgaard, 1964) considers the isotopic fractionation occurring in an isolated air parcel traveling from an oceanic source towards a polar region. The condensed phase is assumed to form in isotopic equilibrium with the surrounding vapor and to be removed immediately from the parcel ( l = 0 in Equation (9)) ... 

Rayleigh model. The three sets of curves are for different initial sea surface temperatures (T ). The approach of Merhvat and Jouzel (1979) is used for the liquid phase, and that of Jouzel and Merlivat (1984) is used for snow formation. The sohd lines correspond to East Antarctic data plotted with respect to either Ts (surface temperature) or 7) (temperature in the atmosphere at the inversion level). 

Figure 1. Lowering of 5 0 by batch decarbonation (straight line) and Rayleigh decarbonation (curves). F is the mole fraction of oxygen remaining in the rock. Note that for Rayleigh decarbonation, tends toward -1000%o if all oxygen is volatilized, but that a calc-silicate limit exists such that F > 0.6 for most metamorphic reactions. There is little difference between the results of the batch and Rayleigh models above F = 0.6 (from Valley 1986).

Two simple models, the Rayleigh model and the steady-state model, are frequently used to interpret N isotope data from the ocean. In both of these models, the degree of consumption of the reactant N pool (/) is a key parameter, and the of the... 

Figure 1 The <5 N of reactant and product N pools of a single unidirectional reaction as a function of the fraction of the initial reactant supply that is left unconsumed, for two different models of reactant supply and consumption, following the approximate equations given in the text. The Rayleigh model (black lines) applies when a closed pool of reactant N is consumed. The steady-state model (gray lines) applies when reactant N is supplied continuously. The same isotopic parameters, an isotope effect e) of 5%o and a <5 N of 5%o for the initial reactant supply, are used for both the Rayleigh and steady-state models, e is approximately equal to the isotopic difference between reactant N and its product (the instantaneous product in the case of the Rayleigh model).

The end-member alternative to the Rayleigh model is the steady-state model, in which reactant N is continuously supplied and partially consumed, with residual reactant N being exported at a steady-state rate such that the gross supply of reactant N equals the sum of the product N and the residual reactant N exported. In this case, the following approximate expessions apply to the reactant N pool (d Nreactant eqn [6]) and the product N pool (d Np oduct eqn [7]) (Figure 1) ... 

Plot the relative scattering of light by the Rayleigh model across the visible spectrum, 400-700 nm. [Note graph is not shown in the answers at the end of this book.]... 

Figure 18. Calculated natural gas plot as a function of the fractional conversion (F) using the Rayleigh model (isotopic fractionation between gas and kerogen at 200°C using the p scission model).

A numerical study is carried out for illustrating the effect of in-stmcture damping models on the optimal distribution of dampers. Models used for the study are the classical Rayleigh model and the non-viscous model given by Equation (1) as it represents the most general damping model within the scope of a linear analysis (Woodhouse 1998). 

Figure 6.9 Thermal conductivity of Mg/PTFE based on either the Maywell or Rayleigh model [33].

---